Weyl modules for equivariant map Lie superalgebras

Abstract

Equivariant map superalgebras are Lie superalgebras of algebraic maps from a scheme to a target finite dimensional Lie superalgebra that are equivariant with respect to the action of a (cyclic) group. In this paper, we extend the notion of Weyl modules, previously defined for the untwisted case, to the case of equivariant(twisted) map superalgebras. Consider $\mathbb{K}$ be an algebraically closed field of characteristic $0$. We define global Weyl modules, and Weyl functors for equivariant map Lie superalgebras $(\g\otimes A)^{\Gamma}$, where $\g$ is a basic classical Lie $\mathbb{K}$-superalgebra and $A$ is an associative commutative unital $\mathbb{K}$-algebra. Under certain assumption on the triangular decomposition of $\g$, we prove that global Weyl modules are universal objects in certain category. We introduce a commutative algebra $\mathbf{A}_{\lambda}^{\Gamma}$ and further prove that global Weyl modules are finitely generated $\mathbf{A}_{\lambda}^{\Gamma}$-modules when $A$ is finitely generated. Finally we define the local Weyl modules for $(\g\otimes A)^{\Gamma}$, where $\g$ is basic classical, using Weyl functors. We show that they are finite dimensional irrespective of the triangular decomposition of $\g^{\Gamma}$. Finally it has been shown that twisted local Weyl modules of $(\g\otimes A)^{\Gamma}$ are precisely the image of the untwisted local Weyl modules under the twisting functor $\textbf{T}_{\mathbf{x}}$.